Hausdorff Distance between Subdifferential sets

Question

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the subdifferentials are bounded, non-empty sets (Theorem 23.4 in Rockafellar). For a given $x$, can we say anything about the Hausdorff distance between $\partial f(x)$ and $\partial g(x)$?

Answer 1

Apparently it is a theorem named Attouch's Theorem proving the convergence of the subdifferentials given some convergence in the functions called epi-convergence.

On the convergence of subdifferentials of convex functions Hedy Attouch, Gerald Beer http://link.springer.com/article/10.1007%2FBF01207197?LI=true

On the convergence of subdifferentials of convex functions Jean-Paul Penot

http://www.sciencedirect.com/science/article/pii/0362546X9390040Y

Don't know if there are results for quantitative convergence though.